Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q9E (page 164)

If, $2 \overset{鈫�}{u} + 3 \overset{鈫�}{v} + 4 \overset{鈫�}{w} = 5 \overset{鈫�}{u} + 6 \overset{鈫�}{v} + 7 \overset{鈫�}{w}$ , then the vectors $\overset{鈫�}{u}, \overset{鈫�}{v}, \overset{鈫�}{w}$ are linearly dependent.

Short Answer

Expert verified

The above statement is true.

If, $2 \overset{鈫�}{u} + 3 \overset{鈫�}{v} + 4 \overset{鈫�}{w} = 5 \overset{鈫�}{u} + 6 \overset{鈫�}{v} + 7 \overset{鈫�}{w}$ then the vectors $\overset{鈫�}{u . \overset{鈫�}{v} \overset{鈫�}{V} .}$ are linearly dependent.

Step by step solution

Definition of linearly dependence vectors

Let V be a vector space over a field K. Let $v_{1}, v_{2}, v_{3}, . . ., v_{n} 鈭� V$ , then if there exist scalars $a_{1}, a_{2}, a_{3}, . . ., a_{n} 鈭� K$ , not all of them 0, such that

$a_{1} v_{1} + a_{2} v_{} + . . . a_{n} v_{n} = 0$

Then the vectors $v_{1}, v_{2}, v_{3}, . . ., v_{n}$ are called linearly dependent vectors.

Otherwise, linearly independent vectors.

To prove whether the vectors are linearly dependent or linearly independent.

We have given,

$2 \overset{鈫�}{u} + 3 \overset{鈫�}{v} + 4 \overset{鈫�}{w} = 5 \overset{鈫�}{u} + 6 \overset{鈫�}{v} + 7 \overset{鈫�}{w}$

$鈬� 3 \overset{鈫�}{u} + 3 \overset{鈫�}{v} + 3 \overset{鈫�}{w} = 0$

Here we can see that the scalars $a_{1}, a_{2} and a_{3}$ are not equal to 0.

i.e. $a_{1} = 3, a_{2} = 3, a_{3} = 3$

Thus, the vectors $\overset{鈫�}{u}, \overset{鈫�}{v}, \overset{鈫�}{w}$ are linearly dependent vectors.

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91影视

If, $2 \overset{鈫�}{u} + 3 \overset{鈫�}{v} + 4 \overset{鈫�}{w} = 5 \overset{鈫�}{u} + 6 \overset{鈫�}{v} + 7 \overset{鈫�}{w}$ , then the vectors $\overset{鈫�}{u}, \overset{鈫�}{v}, \overset{鈫�}{w}$ are linearly dependent.

Short Answer

Step by step solution

Definition of linearly dependence vectors

To prove whether the vectors are linearly dependent or linearly independent.

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91影视

If, 2u鈫�+3v鈫�+4w鈫�=5u鈫�+6v鈫�+7w鈫�, then the vectors u鈫�,v鈫�,w鈫�are linearly dependent.

Short Answer

Step by step solution

Definition of linearly dependence vectors

To prove whether the vectors are linearly dependent or linearly independent.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Logic and Functions

Probability and Statistics

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

If, $2 \overset{鈫�}{u} + 3 \overset{鈫�}{v} + 4 \overset{鈫�}{w} = 5 \overset{鈫�}{u} + 6 \overset{鈫�}{v} + 7 \overset{鈫�}{w}$ , then the vectors $\overset{鈫�}{u}, \overset{鈫�}{v}, \overset{鈫�}{w}$ are linearly dependent.